{
  "id": "1801.02173",
  "version": "v1",
  "published": "2018-01-07T11:44:58.000Z",
  "updated": "2018-01-07T11:44:58.000Z",
  "title": "Weighted estimates for the Calderón commutator",
  "authors": [
    "Jiecheng Chen",
    "Guoen Hu"
  ],
  "comment": "21 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper, the authors establish some weighted estimates for the Calder\\'on commutator defined by \\begin{eqnarray*} &&\\mathcal{C}_{m+1,\\,A}(a_1,\\dots,a_{m};f)(x) &&\\quad={\\rm p.\\,v.}\\,\\int_{\\mathbb{R}}\\frac{P_2(A;\\,x,\\,y)\\prod_{j=1}^m(A_j(x)-A_j(y))}{(x-y)^{m+2}}f(y){\\rm d}y, \\end{eqnarray*} with $P_2(A;\\,x,\\,y)=A(x)-A(y)-A'(y)(x-y)$. Dominating this operator by multi(sub)linear sparse operators, the authors establish the weighted bounds from $L^{p_1}(\\mathbb{R},w_1)$ $\\times\\dots\\times L^{p_m}(\\mathbb{R},w_m)$ to $L^{p}(\\mathbb{R},\\nu_{\\vec{w}})$, with $p_1,\\dots,p_m \\in (1,\\,\\infty)$, $1/p=1/p_1+\\dots+1/p_m$, and $\\vec{w}=(w_1,\\,\\dots,\\,w_m)\\in A_{\\vec{P}}(\\mathbb{R}^{m+1})$. The authors also obtain the weighted weak type endpoint estimates for this operator",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-07T11:44:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weighted estimates",
      "calderón commutator",
      "weighted weak type endpoint estimates",
      "linear sparse operators",
      "calderon commutator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}